None of that tells me (directly) why i (the square root of -1) is so special. You see, finding the square root implies that you've found a number that, when multiplied by itself (2 x 2), equals some other number (4). And that square root (2) is then the solution to the question "what is the square root of this (4)?" But if the number you want to find the square root of is -1 (or any negative number, in fact), it can't have a proper square root because you can't have a negative result when you multiply two negative numbers. You can't get a negative by using one negative and one positive, either; because the definition of squaring a number is multiplying the same number by itself (2 x 2 not 2 x -2). So -1 can't have a square root. Well, it can't under the normal rules we're used to. In comes a concept, invented for this very purpose, which allows there to be roots for negative numbers. Or, at least, a root for -1. That number is called 'i'. There is no "number" for it like "5", "7.987374576576", "789/27" or even "π(pi)". Once you have a square root for -1 (and I'll hope you trust me on this), you can have square roots worked out for any negative number.
There is a catch to having all this power, though. You can have your square root for any negative number, but you can never get rid of i. i becomes a basis for a whole set of numbers called "complex numbers" that tend to look like "23.8678678 + 445.766i". Note the presence of i in the number. You'll get that every time. It's because i is something you can't break down. You can manipulate it, though. That's the whole reason it's around. Math dudes needed a way to get around equations that ran into negative roots. So they dreamt up (actually, rigorously proved) the meaning and existence of i (mathematically speaking). What that allowed them to do is do math with these quantities. So if you run into a negative square root in an equation, you simply get it to factor out (which means making it disappear in the mathematical sense) using i (if you can) and then move on. When it was just a negative square root with no context you couldn't do that.
So WHAT??? =]
All this got me to thinking about how it related to things I see in software. I was just training on new products we have (new to me anyway), and we ran into an odd thing. The product can retrieve the structure of queries running in a database; this code is called SQL - Structured Query Language. Well, this SQL code can get quite odd and databases made by IBM don't have the same flavor of oddness as databases made by Microsoft or whomever. We were running the tool against Microsoft's Database (SQL Server 2000 sp3a for the curious), and I got different (incorrect) results than the trainer. After a moments thought, she said "Oh yes, you need to change a setting to get SQL with double quote(") characters in it".
Later on, sitting on this airplane, I was reading about the Riemann Hypothesis (a major mathematical theory). And the author was doing what all good pop-sci and pop-math authors do by giving a refresher for those of us who are not math professionals. He went over the different number families (real numbers, rational numbers, irrational numbers, complex numbers, etc.). And he gave a good explanation of i (somewhat abridged here). What struck me was that he said that i was invented out of need for people to get past restrictions in some equations dealing with negative numbers needing to have square roots. Well, that seemed very like my problem with the software.
I needed to have the software get the SQL from the database. The mathematicians needed to have their equations get the solutions at certain values. In the software problem, there was a condition in the database (the use of the double quotes) which requires the invention of a novel and narrowly applicable solution (the setting in our software) to get over the condition. In the math world, they invent a new number, i, to allow them to deal with the limit in the numbers they have in the context of the equations they'd like to solve. In the software case, the whole thing is a situation wholly invented by people. Math (and the following point is certainly debated but I give my own bias here) is a tool that describes the ideal forms of the world around us - the reduced world. The interesting question for me (and the point of this diatribe) is this:
The software problem existed wholly due to the way the systems were built and approached and some fore-thought on either end would remove the need for the particular solution used. Could there be some more fundamental shift in thought hiding in the number i that would make the need for it go away as well?
In other words, is i just a handy way of dealing with something that is too messy to treat properly? I tend to think in a systems view. Math, the software, the world are all systems. Those systems are different in details for sure, but, as they are all systems, share some general properties with all systems. Whenever we invent some great leap of faith, or sum up some complex idea with a small symbol, are we just shoving some complexity into a neat hidey-hole so we needn't see it anymore instead of tackling head on? Now, to be fair, there have been many man-years spent trying to tackle the nature of i (all puns welcome). My question is more to the nature of solutions that are there merely to be a pass through. No mathematician actually deals with i. They just use it to get rid of something they don't like and get rid of it (i) in the process. It would be as if, finding a quality I didn't like in my dog, I simply trained my dog not to have that quality through using it against the dog (I teach him not to howl at the moon by playing tapes of his howling at the moon to him on a 24 hour loop until he stops). Those solutions don't deal with how the problem arises in the first place.
It doesn't get at the root of the issue (puns again, sorry).
What is the way one deals with these persistent and seemingly unsolvable problems? Don't they point to some lack of deeper understanding? Or are they the sign that our blind grasping at the bounds of knowledge are making contact with the walls in the dark? Are we feeling our way around the room we're bound inside by our minds and senses; only knowing we've struck the wall when these problems hit? And, of course, all we can do is repel off the wall using it's own strength so we can float about the middle of the room again to await the next time we hit a wall. What if we punched the wall all the harder? Or is that a metaphorical number i, trying to solve this issue by simply using mysticism against a mystery? hmmm...
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